Thurston

The Mathematics

William Thurston conjectured that three-dimensional spaces are built from eight fundamental geometries — familiar ones like Euclidean and hyperbolic space, alongside stranger ones that twist and shear. This collaborative work gives intrinsic, first-person views of these spaces, and of compact manifolds built from them. The image here shows the Seifert–Weber space, with the edges of its dodecahedral fundamental domain visible in infinite repetition as light wraps around the manifold.

Technique

We wrote the skeleton of a ray-tracing program in general geometric language — points, tangent vectors, distances, isometries, quotient spaces, geodesics, parallel transport. We then supplied an implementation of each piece for all eight Thurston geometries, allowing real-time rendering of perspectively correct inside views.